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QUANTUM WEYL ALGEBRAS and DEFORMATIONS of U(G) Pacific Journal of Mathematics QUANTUM WEYL ALGEBRAS AND DEFORMATIONS OF U.g/ NAIHUAN JING AND JAMES ZHANG Volume 171 No. 2 December 1995 PACIFIC JOURNAL OF MATHEMATICS Vol. 171, No. 2, 1995 QUANTUM WEYL ALGEBRAS AND DEFORMATIONS OF U(G) NAIHUAN JING AND JAMES ZHANG We construct new deformations of the universal enveloping algebras from the quantum Weyl algebras for any R-matrix. Our new algebra (in the case of g = SI2) is a noncommuta- tive and noncocommutative bialgebra (i.e. quantum semi- group) with its localization being a Hopf algebra (i.e. quan- tum group). The ring structure and representation theory of our algebra are studied in the case of SI2. Introduction. Quantum groups are usually considered to be examples of ^-deformations of universal enveloping algebras of simple Lie algebras or their restricted dual algebras. The simplest example is the Drinfeld-Jimbo quantum group λ C/g(sl2), which is an associative algebra over C(q) generated by e,/, k,k~ subject to the following relations: ke = q2ek, kf = q~2fk, k-k-1 ef-fe =q-q-1 One sees that the element k does not stay in the same level as the elements e and / do, since k is actually an exponential of the original element h in the Cartan subalgebra. There do exist deformations to deform all generators into the same level, for example, Sklyanin algebras. However it is still uncertain if Sklyanin algebra is a bialgebra or not. Using the quantum Weyl algebras discussed in [WZ, GZ], we will con- struct new deformations for the enveloping algebras which have such a nice property. Our algebra in the case of sl2 is an associative algebra generated by e, /, h with the relations: qhe — eh — 2e, hf - qfh = -2/, ef-qfe^h + ^h2. 437 438 NAIHUAN JING & JAMES ZHANG Moreover the algebra is a bialgebra with the comultiplication A(x) = x ® 1 + 1 ® x + -(1 - q)h ® a; with x being any of the basic generators e, /, Λ. Its localization is a Hopf algebra. Clearly our algebra is noncommutative and noncocommutative or a quantum group according to Drinfeld. Our new deformations will enjoy all the nice properties carried by the enveloping algebras such as having the same global, Krull, Gelfand-Kirillov dimensions as well as the same Hubert series for the associated graded rings. Our deformation is similar to Witten's algebra ([W], see also [L]) in one aspect that the Casimir element in our algebra skew-commutes with other generators. We also remark that the homogenization of both Witten's alge- bra and our algebra belong to a single family of algebras studied in a recent work [LSV] from a different perspective. The idea behind our construction is very simple and natural. We start from the derivatives and multiplication operators on a skew polynomial ring associated to a Hecke R-matrix. They form a quantum Weyl algebra as discussed by [GZ]. Then we go on to consider the subalgebra generated by the quadratic elements to obtain a new deformation. We study the ring structure for our new algebras in the case of sl2. The representation theory is also worked out for both generic and singular cases of q. It is not surprising that the representation theory of the new algebra is quite similar to that of the Drinfeld-Jimbo quantum algebra Uq(g). Our approach is general in the sense that our method works for a wide class of R-matrices. In other words, we just reverified the profound principle that a meaningful R-matrix can give a deformation to the simple Lie algebras [FRT]. Recently Ding and Prenkel [DF] have introduced another kind of quantum Weyl and Clifford algebras from the L-operator approach of the quantum inverse scattering method and use them to give representations of the quan- tum algebras. Their work differs from ours in forming a different quadratic expression for the basic generators to give realization of the usual Drinfeld- Jimbo quantum algebras, while our result is to focus on obtaining a new quantum algebra structure. We will use the usual symbol Uq(g) to denote our new deformed algebra unless indicated otherwise. NEW DEFORMATIONS 439 1. Quantum Weyl Algebras. Let R = (rfj) be a Hecke R-matrix. Namely, R is a n2 by n2 matrix satisfying the Hecke relation and braid relation: 1 2 ι (1.1) (R-q)(R + q- )=0 or R = 9Λ - q~ R + 1; The braid relation (1.2) means that for all i,j,g and all u,k,f kf ul st (Λ *\\ V r r r — Y^ r*t uk If \L'°) Z_^rlt 'is' jg — Z^'ij'sl ' tg l,s,t l,s,t summed over the repeated indices Z,s,£ and here i? = (rfj). Given such an R-matrix, we can define the quantum Weyl algebra An(R) as follows. Quantum Weyl algebras were first considered in [WZ]. Here we follow [GZ] to define the quantum Weyl algebra An(R). The generators are 1 n ι xl5 , xn and 9 , , 9 , and each d corresponds to the partial derivative in the ordinary Weyl algebra. The relations are (1.4.1) (1.4.2) &Xj •• (1.4.3) S,t for all i and j. We also assume that R has a skew inverse matrix in the following sense: there is P = (pf*) such that ΣpfMϊ = <5ifc<^u = Σriίpί ϊj which is always true in the considered cases. If such P did not exist then ι ι we could not write xkd in terms of d Xj as we see from (1.4). We refer the reader to [GZ] for a detailed explanation why the relations (1.4) are natural analog of the usual defining relations for the classical Weyl algebra realized on the polynomial algebra in n variables. In the classical case, there is a well-known homomorphism from the en- j veloping algebra C/(gln) to the n-th Weyl algebra An by sending ei3 to Xid . (This homomorphism is not injective.) Motivated by this classical fact we are going to construct a quantum enveloping algebra generated by the quan- tum elements XidK Let's denote the element x$ by e^, then we have the following commutation relations for e\. Proposition 1.1. For all n, z, /, k 440 NAIHUAN JING & JAMES ZHANG (b) = Σ rfc + Q-1 Σ (Σ tf βί) (Σ rteή - <τx Σ 4 (Σ » Proof. We only give a proof for (a) (and (b) can be handled similarly). It follows from (1.4.2) that ι X& xkd = Xi Then we have (Σ r"ίe0 • 4 = Σ '.I^+1Σ r^^ = Σ * 2W + Σr:;^ fa*' = Σ ^"^i^ + Σ rTjήtr^x^d'd1 by (1.4.1) = Σ r?k*i& + Σ rϋrj^x^d'd1 by (1.2) >^ + Σ ήlr^Xvq-1 (Vx, - Jj) ^ by (1.4.2) = Σ r%*i& + I"' Σ r&rZxv#z.& - = Σ rSeJ + 4-1 Σ (Σ C<) (Σ rίW. D Remark. We conjecture that the relations in (a) are equivalent to the relations in (b). For simplicity we introduce some notations: u 1 u u u Of course, by (1.4.2), b z = q' (d xz - δ z). Then the matrix (b z)nxn is denoted by B. cul = V^ ruieι- = V^r^x θ' and the matrix (c^.)nxn is denoted by C\ It is easy to see that ΣC% = B. zk / j zi k / j NEW DEFORMATIONS 441 and the matrix (d^)nxn is denoted by D\. It is easy to see that ΣD* = B. By using these notations we can re-write the relations in (a) and (b) as follows: u ι 2 ul ι u ι ι 1 u (a) h ep — π~ Lr —n- fi pe 4- π~ V r^c hυ W °z k ~ H zk H °z k ^ Q λ^ zk j for which we use R2 = (q — q~ι)R + 1. In the matrix form, where / is the identity matrix. (b) 4K = q-2d% ~ q-'δ^A + q-ι ΣC^, or, e{B = q-'BDf + q-2D{ - q^Ie?. By (a) By(b) = q~xBB + q~2B - q~xI • T. As a consequence B • T = T • B and (B + q~γT){B - qT) = 0. Corollary 1.2. The element T = Σe\ commutes with elements xkd{ for all k,l in the quantum Weyl algebra An{R). Proof. This follows from the equality TB = BT and the fact that e\ = 2. Definition of C/,(gln). We are going to construct a deformation of the enveloping algebra t/(gln) out of the subalgebra generated by the quadratic elements Xid}. We can not use all the relations in Proposition 1.1 (a), since in the classical case, the map from £7(gln) or ί7(sln) for n > 2 to the n-th Weyl algebra An by 442 NAIHUAN JING & JAMES ZHANG sending e^ to Xidj is not injective. We need to find out what are the proper relations for Uq(gln). Let R be the R-matrix for the standard quantum group GLn(q). As a tensor, we may write R as As a matrix R is given by R = (r*j) and rjj = 9, Γ ) = 1 for all i, j, and r*j = # — g"1 for all i < j. The quantum Weyl algebra constructed from this particular R-matrix is denoted by An(q). It follows from (1.4) in Section 1 that the relations of An(q) are the fol- lowing: (2.1.1) XiXj = qXjXi,Vi < j; (2.1.2) d'dj =ιj (2.1.3) drx^ 2 l 2 j (2.1.4) d'xi = 1 + q Xid + (q - 1) Σx3d ,ML If we denote \ij\ = 1 for i < j, \ij\ = 0 for i = j, and |ΐj| = —1 for i > j, then (2.1.1) and (2.1.2) can be re-written as (2.1.1') xixj = qMχjxiyiJ, (2.1.27) d'dj = q-liJldjd\Vi,j.
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